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On Ramanujan’s Nested Radicals

Friday, December 30th, 2011 08:50 / 8 Comments

Ramanujan (1887-1920) discovered some formulas on algebraic nested radicals. This article is based on one of those formulas. The main aim of this article is to discuss and derive them intuitively. Nested radicals have many applications in Number Theory as well as in Numerical Methods.
batman-3

How to Draw the Famous Batman Curve

Saturday, September 24th, 2011 16:41 / 3 Comments

The ellipse \displaystyle \left( \frac{x}{7} \right)^{2} + \left( \frac{y}{3} \right)^{2} - 1 = 0 looks like this:

ellipse

So the curve \left( \frac{x}{7} \right)^{2}\sqrt{\frac{\left| \left| x \right|-3 \right|}{\left| x \right|-3}} + \left( \frac{y}{3} \right)^{2}\sqrt{\frac{\left| y+3\frac{\sqrt{33}}{7} \right|}{y+3\frac{\sqrt{33}}{7}}} - 1 = 0 is the above ellipse, in the region where |x|>3 and y > -3\sqrt{33}/7:

ellipse cut

That’s the first factor.
(more…)

A Problem (and Solution) from Bhaskaracharya’s Lilavati

Saturday, March 5th, 2011 08:43 / 4 Comments

I was reading a book on ancient mathematics problems from Indian mathematicians. Here I wish to share one problem from Bhaskaracharya‘s famous creation Lilavati. (more…)

ramanujan

Solving Ramanujan’s Puzzling Problem

Thursday, February 3rd, 2011 14:07 / 11 Comments

Consider a sequence of functions as follows:-

f_1 (x) = \sqrt {1+\sqrt {x} }
f_2 (x) = \sqrt{1+ \sqrt {1+2\sqrt {x} } }

f_3 (x) = \sqrt {1+ \sqrt {1+2 \sqrt {1+3 \sqrt {x} } } }

……and so on to

f_n (x) = \sqrt {1+\sqrt{1+2 \sqrt {1+3 \sqrt {\ldots \sqrt {1+n \sqrt {x} } } } } }

Evaluate this function as n tends to infinity.

Or logically:

Find

\displaystyle{\lim_{n \to \infty}} f_n (x) .

(more…)

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